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Algebras of Holomorphic Functions and Control Theory

Dover Publications, Inc.

Contents

Control theory

The basic objects of study in control theory are underdetermined differential /difference equations. By underdetermined, we mean that the functions in the differential equation system are not uniquely determined, but some of them are 'free', that is, they can be arbitrarily chosen (something that we can 'input'). Once this choice is made, then the rest of the variables are uniquely determined.

The basic question in control theory is: Can one influence the behaviour of some of the functions in the system by appropriate manipulation of the free variables, that is, can one control the behaviour of the system? We elaborate on this below.

For instance, the algebraic equation x = 10 - u, where x, u are integers, is underdetermined. Indeed, it is not the case that there is a unique pair of integers (x, u) that satisfies the equation. Viewing the variable u as free (something we can decide, or 'input'), we see that the x is then determined via x = 10 - u. One could then ask the question of manipulating the to-be-controlled variable x by suitable changing the input u. For example, in order to make x< 5, we can do so by inputting a u > 5! In control theory, one addresses a similar question of control, but for underdetermined difference or differential equations.

Our basic object of study in control theory will be the system

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where n = 0, 1, 2, 3, ... serves as the time axis. Here the variable

x is called the state,

u is called the input,

y is called the output.

The state x at any time n is an element belonging to a Hilbert space X, called the state space. On the other hand, u and y are complex-valued sequences. The symbols A, B, C, D denote continuous linear transformations between appropriate spaces, and are known from the modelling procedure:

A [member of] L(X),

B [member of] L(C, X)

C [member of] L(X, C),

D [member of] L(C) = C.

If we look at the first of the two equations in (1.2), then we notice that this is an underdetermined equation, in the sense that the input u can be chosen freely, that is, it can be specified arbitrarily. It is something that we can choose, or input into the system. Once such a choice of the input u has been made, then, given the initial condition x(0), the first equation in (1.2) determines the state x uniquely. Indeed, if we decide what u(0), u(1), u(2), u(3), ... are, then with x(0) known, we see that x(1), x(2), x(3), ... are determined:

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Now that x is determined, the output y is then determined by the second equation in (1.2):

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Thus we can think of the system described by (1.2) as a box which given the input u, manufactures the output y. See Figure 1.

One may then ask the question about 'controlling' y, that is, manipulating the behaviour of the output in some desired manner, by suitably changing the input u. For this reason (that is, there may exist a possibility of controlling the behaviour of the system), a system described by an equation of the form (1.2) is called a control system.

1.1. Finite vs infinite; discrete vs continuous

The control system (1.2) is called finite-dimensional if X is finite-dimensional, while it is called infinite-dimensional if X is infinite-dimensional.

Note that we have considered the time axis to be discrete. For this reason, we call our control systems as discrete-time control systems. One may also consider similar underdetermined differential equation models, with a continuous time axis:

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(1.1)

A model described by a system of high order linear ordinary differential equations with constant coefficients can be expressed as a first order equation (1.1) with an appropriate choice of the state variable x, and state space X = In.

On the other hand a model described by a system of linear partial differential equations or ordinary differential equations with delays can also be rewritten as (1.1), but now one needs an infinite-dimensional Hilbert space X as the state space.

We will not go into the details of these modelling issues in this monograph, since there are several good sources on the subject available to the interested reader; see for instance [CZ].

Throughout this monograph, we will only consider discrete-time systems although an analogous theory exists for continuous-time systems as well. Moreover, we will primarily be interested in infinite-dimensional systems, since only there do the subtle algebraic properties of the rings of analytic functions arise in the control theoretic problem of stabilization, as we shall soon see. As opposed to this, in the finite-dimensional case, a much more complete theory is available; see for example [V].

1.2. Output in terms of the input

In the rest of this book, we will consider the discrete-time control system

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(1.2)

where at any time n, x(n) belongs to a Hilbert space X, u(n), y(n) belong to I, and where A [member of] L(X), B [member of] (I, X), C [member of] (X, I) and D [member of] L(I) = I

The output y can be expressed in terms of the input u, the initial condition x(0), and the 'system parameters' A, B, C, D as follows:

Proposition 1.1.Let u = (u(n))n ≥ 0 be a sequence and x(0) [member of] X. Then the state x and the output y to (1.2) is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(1.3)

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(1.4)

for n = 1,2,3,....

Proof. (1.4) follows immediately from (1.3), and we prove the latter by induction on n.

The case n = 1 follows from the state equation

x(1) = Ax(0) + Bu(0).

If (1.3) holds for some n, then we have

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This completes the proof. ?

1.3. Transfer function

The relation (1.4) can be expressed in yet another way by means of the 'z - transform'. Roughly speaking, the z-transform takes a sequence to a holomorphic function. We give the precise definition below. It is via this z-transform that holomorphic functions make an appearance in control theory.

Definition 1.2. Let H be a Hilbert space. If h = (h(n))n ≥ 0 denotes a H-valued sequence, then its z-transform is the holomorphic function [??] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(1.5)

which is defined for all z in a disk with center 0 and radius R in [0, +∞], given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(1.6)

It can be shown that if |z| < R, the series (1.5) converges absolutely. If |z| >R, the series diverges. If 0 < r< R, then the series converges uniformly on {z [member of] I||z| ≤ r}.

The number R given by (1.6) is called the radius of convergence of the power series (1.5).

We also note that the radius of convergence of (1.5) is positive when the sequence h is exponentially bounded, that is, there exist M, α > 0 such that

||h(n)|| ≤ Mαn, n = 0, 1, 2, 3,....

Indeed, we then have

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and so

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All this is probably familiar to the student from complex analysis for the usual complex-valued sequences, but although the above sequence is vector-valued, the same proofs work, mutatis mutandis.

Now suppose that the initial condition is x(0) = 0.

When the initial condition is x(0) = 0, it turns out that the z – transforms of the output and input are related in a particularly simple manner, namely by multiplication by a special function g, called the transfer function of the system. More precisely, we will soon show that by taking z-transform in (1.2), we obtain the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for z in a sufficiently small disk (which guarantees that

1 [not member of] zσ(A) := {zλ | λ [member of] σ(A)},

where σ(A) denotes the spectrum of A). Indeed the spectrum of a continuous linear transformation A [member of] L (X) is a bounded set, since if |λ| > ||A||, then

λI - A = λ (I - 1/λ A)

is invertible in L(X), and so σ(A) [subset] {λ [member of] I| |λ| ≤ ||A||}. So for a small enough disk Δ around 0, we have that 1 [not member of] zσ(A) for all z lying in Δ, and so I - zA is invertible in L(X) for such z.
(Continues...)

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Excerpted from Algebras of Holomorphic Functions and Control Theory by Amol Sasane. Copyright © 2014 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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